Physics Library
 An open source physics library
Encyclopedia | Forums | Docs | Random | Template Test |  
Login
create new user
Username:
Password:
forget your password?
Main Menu
Sections

Talkback

Downloads

Information
thin equivalence relation (Definition)

Thin equivalence relation

Definition 1.1  

Let $a,a' : x \simeq y $ be paths in $X $. Then $a$ is thinly equivalent to $a' $, denoted $a \sim_{T} a' $, if there is a thin relative homotopy between $a $ and $a' $.

We note that $\sim_{T} $ is an equivalence relation, see [2]. We use $\langle a \rangle : x \simeq y $ to denote the $\sim_{T} $ class of a path $a: x \simeq y $ and call $\langle a \rangle $ the semitrack of $a $. The groupoid structure of $\boldsymbol{\rho}^\square_1 (X) $ is induced by concatenation, +, of paths. Here one makes use of the fact that if $a: x \simeq x', \ a' : x' \simeq x'', \ a'' : x'' \simeq x''' $ are paths then there are canonical thin relative homotopies

\begin{displaymath} \begin{array}{r} (a+a') + a'' \simeq a+ (a' +a'') : x \simeq... ...) \simeq e_{x} : x \simeq x \ ({\it cancellation}). \end{array}\end{displaymath}

The source and target maps of $\boldsymbol{\rho}^\square_1 (X)$ are given by

$\displaystyle \partial^{-}_{1} \langle a\rangle =x,\enskip \partial^{+}_{1} \langle a\rangle =y,$
if $\langle a\rangle :x\simeq y$ is a semitrack. Identities and inverses are given by

$\displaystyle \varepsilon (x)=\langle e_x\rangle \quad \mathrm{ resp.} -\langle a\rangle =\langle -a \rangle.$

Bibliography

1
K.A. Hardie, K.H. Kamps and R.W. Kieboom, A homotopy 2-groupoid of a Hausdorff space, Applied Cat. Structures, 8 (2000): 209-234.
2
R. Brown, K.A. Hardie, K.H. Kamps and T. Porter, A homotopy double groupoid of a Hausdorff space, Theory and Applications of Categories 10,(2002): 71-93.



"thin equivalence relation" is owned by bci1.

View style:

Keywords:  thin equivalence relation

Cross-references: identities, target maps, groupoid, equivalence relation, homotopy, thinly equivalent

This is version 1 of thin equivalence relation, born on 2009-04-19.
Object id is 670, canonical name is ThinEquivalenceRelation.
Accessed 292 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:

No messages.

Testing some escape charachters for html category with a generator has an injective cogenerator" now escape ” with "